q-deformed Fock spaces of level one. The product theorem is regarded as a formal q-analogue of the tensor product theorem of level one. In this talk, we show a higher level analogue of Leclerc-Thibon product theorem under a suitable multi charge condition.

In terms of Bruhat graphs, we establish combinatorial inequalities on q=1 values of KL polynomials for crystallographic Coxeter systems: Based on the author's recent result on the existence of a strict inequality of KL polynomials, we give a lower bound of their q=1 values by graph-theoretic distance.